This chapter proposed a class of LR-NSV models with generalized Student’s t-error distributions defined by applying three families of power transformation—
exponential, modulus, and Yeo–Johnson—to lagged log volatility. We developed an efficient MCMC algorithm including Gibbs, HMC, and RMHMC steps for sam-pling from the posterior of the models. Empirical results using TOPIX data show that the posterior mean accepts all proposed models and the 90% HPD interval only accepts some of the proposed models. Hence, these results provide evidence in support of the LR-NSV models. Furthermore, the marginal likelihood and Bayes factor criterion indicate that the proposed LR-NSV model outperforms the LRSV model, where the LR-MTSV models best fitted the returns data having a very high kurtosis and worst fitted the data having a small kurtosis. In addition, the LR-NSV model with SKT distribution performs the best among the four returns distribution specifications considered in this chapter. Finally, the LR-MTSV-SKT model’s performance has been considered under three alternative prior distribution assumptions for the power parameter. The results show considerable robustness for priors with very diffused distributional behaviour.
The proposed LR-NSV models could also be extended by considering a class of power transformations for RV.Gon¸calves and Meddahi(2011) particularly showed that the log transformation of RV can be improved upon by choosing values for the BC parameter other than zero. Another possible extension to the multivariate R-NSV is suggested because procedures for extensions of the RSV model and power transformations, especially the BC transformation, to multivariate data have been proposed in previous studies.
Appendices
5.A Posterior Summary Statistics of the Power Parameter
Table 5.5: Summary of the posterior sample of the power parameter λin the LR-NSV and LR-LR-NSV-T models for the TOPIX 2004–2007 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
LR-NSV model λET
Mean (SD) −0.039 (0.021) −0.037 (0.020) −0.051 (0.022) −0.038 (0.021) 90% HPD [−0.072,−0.003] [−0.069,−0.001] [−0.086,−0.011] [−0.072,−0.003]
IF (NSE) 3.6 (0.0004) 3.8 (0.0004) 3.5 (0.0004) 4.0 (0.0004) λM T
Mean (SD) 0.958 (0.063) 0.969 (0.068) 0.943 (0.060) 0.985 (0.066) 90% HPD [0.857,1.061] [0.856,1.075] [0.846,1.037] [0.871,1.085]
IF (NSE) 21.9 (0.0032) 32.3 (0.0036) 22.7 (0.0031) 23.4 (0.0029) λY J
Mean (SD) 0.941 (0.030) 0.941 (0.030) 0.921 (0.032) 0.939 (0.030) 90% HPD [0.890,0.990] [0.892,0.990] [0.868,0.975] [0.889,0.989]
IF (NSE) 3.2 (0.0006) 3.4 (0.0005) 3.2 (0.0006) 3.5 (0.0005) LR-NSV-T model
λET
Mean (SD) −0.037 (0.021) −0.035 (0.020) −0.050 (0.023) −0.036 (0.020) 90% HPD [−0.073,−0.004] [−0.071,−0.003] [−0.089,−0.013] [−0.070,−0.002]
IF (NSE) 4.1 (0.0003) 4.3 (0.0003) 4.5 (0.0005) 5.1 (0.0005) λM T
Mean (SD) 0.981 (0.068) 0.997 (0.061) 0.981 (0.069) 1.006 (0.066) 90% HPD [0.874,1.094] [0.891,1.091] [0.871,1.095] [0.893,1.108]
IF (NSE) 32.9 (0.0036) 20.1 (0.0026) 35.5 (0.0045) 23.5 (0.0035) λY J
Mean (SD) 0.943 (0.030) 0.943 (0.030) 0.923 (0.032) 0.941 (0.031) 90% HPD [0.894,0.993] [0.891,0.991] [0.872,0.977] [0.891,0.993]
IF (NSE) 3.4 (0.0006) 3.9 (0.0006) 3.7 (0.0007) 4.2 (0.0007)
Table 5.6: Summary of the posterior sample of the power parameter λin the LR-NSV-NCT and LR-NSV-SKT models for the TOPIX 2004–2007 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
LR-NSV-NCT model λET
Mean (SD) −0.037 (0.020) −0.036 (0.020) −0.048 (0.022) −0.037 (0.021) 90% HPD [−0.071,−0.005] [−0.070,−0.002] [−0.085,−0.010] [−0.073,−0.002]
IF (NSE) 3.9 (0.0004) 4.8 (0.0004) 5.1 (0.0005) 4.3 (0.0005) λM T
Mean (SD) 0.981 (0.062) 0.987 (0.063) 0.986 (0.063) 1.010 (0.063) 90% HPD [0.878,1.077] [0.883,1.089] [0.887,1.090] [0.900,1.111]
IF (NSE) 26.5 (0.0029) 22.2 (0.0026) 24.3 (0.0032) 21.6 (0.0026) λY J
Mean (SD) 0.942 (0.029) 0.944 (0.030) 0.923 (0.032) 0.941 (0.030) 90% HPD [0.894,0.990] [0.895,0.996] [0.872,0.978] [0.891,0.993]
IF (NSE) 3.5 (0.0006) 3.9 (0.0006) 3.9 (0.0007) 3.8 (0.0006) LR-NSV-SKT model
λET
Mean (SD) −0.037 (0.020) −0.036 (0.021) −0.050 (0.023) −0.035 (0.020) 90% HPD [−0.072,−0.004] [−0.070,−0.002] [−0.087,−0.010] [−0.068,−0.002]
IF (NSE) 4.9 (0.0004) 5.3 (0.0005) 4.9 (0.0005) 5.2 (0.0004) λM T
Mean (SD) 0.992 (0.063) 0.996 (0.064) 0.983 (0.062) 1.016 (0.062) 90% HPD [0.889,1.094] [0.890,1.099] [0.883,1.083] [0.906,1.112]
IF (NSE) 26.9 (0.0036) 29.1 (0.0038) 21.8 (0.0022) 21.2 (0.0023) λY J
Mean (SD) 0.942 (0.029) 0.944 (0.031) 0.923 (0.031) 0.941 (0.030) 90% HPD [0.896,0.993] [0.893,0.995] [0.871,0.975] [0.893,0.992]
IF (NSE) 3.5 (0.0005) 3.9 (0.0005) 3.4 (0.0006) 4.4 (0.0005)
Table 5.7: Summary of the posterior sample of the power parameter λin the LR-NSV and LR-LR-NSV-T models for the TOPIX 2004–2011 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
LR-NSV model λET
Mean (SD) 0.015 (0.008) 0.011 (0.008) 0.014 (0.009) 0.011 (0.008) 90% HPD [0.000,0.029] [−0.003,0.025] [−0.001,0.029] [−0.002,0.025]
IF 3.5 (0.0001) 3.1 (0.0001) 3.2 (0.0001) 2.7 (0.0001) λM T
Mean (SD) 1.054 (0.026) 1.054 (0.029) 1.042 (0.027) 1.057 (0.028) 90% HPD [1.0104,1.0974] [1.004,1.101] [0.997,1.087] [1.011,1.107]
IF (NSE) 12.95 (0.0007) 16.5 (0.0011) 13.6 (0.0009) 14.2 (0.0009) λY J
Mean (SD) 1.025 (0.014) 1.018 (0.014) 1.022 (0.015) 1.018 (0.014) 90% HPD [1.001,1.049] [0.995,1.042] [0.996,1.048] [0.995,1.042]
IF (NSE) 3.0 (0.0002) 2.8 (0.0002) 3.8 (0.0003) 2.7 (0.0002) LR-NSV-T model
λET
Mean (SD) 0.016 (0.008) 0.011 (0.008) 0.014 (0.008) 0.011 (0.008) 90% HPD [0.002,0.030] [−0.002,0.025] [0.000,0.029] [−0.001,0.024]
IF (NSE) 3.1 (0.0001) 2.9 (0.0001) 3.2 (0.0001) 2.4 (0.0001) λM T
Mean (SD) 1.054 (0.028) 1.052 (0.029) 1.043 (0.028) 1.058 (0.029) 90% HPD [1.006,1.099] [1.004,1.102] [0.996,1.089] [1.008,1.106]
IF (NSE) 15.2 (0.0010) 14.3 (0.0011) 15.1 (0.0010) 15.7 (0.0011) λY J
Mean (SD) 1.025 (0.014) 1.018 (0.014) 1.022 (0.015) 1.018 (0.013) 90% HPD [1.002,1.049] [0.994,1.042] [0.996,1.047] [0.996,1.040]
IF (NSE) 2.9 (0.0002) 2.9 (0.0002) 3.5 (0.0002) 2.7 (0.0002)
Table 5.8: Summary of the posterior sample of the power parameter λin the LR-NSV-NCT and LR-NSV-SKT models for the TOPIX 2004–2011 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
LR-NSV-NCT model λET
Mean (SD) 0.016 (0.008) 0.011 (0.008) 0.014 (0.009) 0.012 (0.008) 90% HPD [0.002,0.030] [−0.001,0.025] [0.000,0.029] [−0.001,0.025]
IF (NSE) 3.3 (0.0001) 2.5 (0.0001) 2.9 (0.0001) 2.4 (0.0001) λM T
Mean (SD) 1.054 (0.028) 1.052 (0.029) 1.044 (0.028) 1.060 (0.028) 90% HPD [1.008,1.101] [1.006,1.104] [0.999,1.093] [1.011,1.105]
IF (NSE) 13.3 (0.0011) 14.7 (0.0011) 13.5 (0.0010) 17.3 (0.0010) λY J
Mean (SD) 1.025 (0.015) 1.018 (0.014) 1.022 (0.015) 1.017 (0.014) 90% HPD [1.001,1.051] [0.994,1.041] [0.996,1.047] [0.994,1.040]
IF (NSE) 3.3 (0.0002) 2.8 (0.0002) 3.5 (0.0003) 2.5 (0.0002) LR-NSV-SKT model
λET
Mean (SD) 0.016 (0.008) 0.012 (0.008) 0.014 (0.008) 0.011 (0.008) 90% HPD [0.002,0.029] [−0.001,0.025] [0.000,0.028] [−0.001,0.025]
IF (NSE) 2.9 (0.0001) 2.8 (0.0001) 2.7 (0.0001) 2.8 (0.0001) λM T
Mean (SD) 1.056 (0.027) 1.055 (0.030) 1.044 (0.028) 1.061 (0.030) 90% HPD [1.012,1.102] [1.006,1.107] [0.999,1.095] [1.011,1.109]
IF (NSE) 12.8 (0.0009) 18.2 (0.0012) 14.1 (0.0010) 14.4 (0.0012) λY J
Mean (SD) 1.026 (0.014) 1.018 (0.013) 1.022 (0.015) 1.018 (0.013) 90% HPD [1.003,1.050] [0.995,1.041] [0.998,1.047] [0.996,1.040]
IF (NSE) 3.1 (0.0002) 2.6 (0.0002) 3.2 (0.0002) 2.6 (0.0002)
5.B Posterior Summary Statistics of the Persis-tence of Volatility
Table 5.9: Summary of the posterior sample of the persistence of log volatility, φ, in the LR-NSV and LR-NSV-T models for the TOPIX 2004–2007 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
LR-NSV model φET
Mean (SD) 0.941 (0.010) 0.933 (0.012) 0.941 (0.011) 0.931 (0.013) 90% HPD [0.925,0.960] [0.912,0.953] [0.923,0.959] [0.909,0.952]
IF (NSE) 7.28 (0.0003) 11.4 (0.0004) 6.4 (0.0003) 9.1 (0.0004) φM T
Mean (SD) 0.962 (0.024) 0.952 (0.028) 0.968 (0.022) 0.945 (0.028) 90% HPD [0.9270,0.9993] [0.913,0.998] [0.937,0.999] [0.903,0.994]
IF (NSE) 25.83 (0.0013) 33.0 (0.0014) 24.6 (0.0012) 27.3 (0.0014) φY J
Mean (SD) 0.941 (0.011) 0.932 (0.012) 0.940 (0.011) 0.930 (0.012) 90% HPD [0.922,0.958] [0.911,0.951] [0.921,0.959] [0.910,0.952]
IF (NSE) 7.6 (0.0003) 8.5 (0.0003) 9.5 (0.0004) 8.5 (0.0003) LR-NSV-T model
φET
Mean (SD) 0.940 (0.011) 0.932 (0.012) 0.938 (0.012) 0.929 (0.013) 90% HPD [0.922,0.959] [0.911,0.953] [0.919,0.958] [0.907,0.951]
IF (NSE) 7.9 (0.0003) 9.6 (0.0004) 7.9 (0.0003) 9.6 (0.0005) φM T
Mean (SD) 0.953 (0.027) 0.943 (0.026) 0.955 (0.026) 0.938 (0.029) 90% HPD [0.915,0.997] [0.902,0.988] [0.916,0.997] [0.893,0.988]
IF (NSE) 32.6 (0.0014) 21.4 (0.0010) 39.2 (0.0018) 25.3 (0.0016) φY J
Mean (SD) 0.940 (0.011) 0.932 (0.012) 0.937 (0.011) 0.929 (0.013) 90% HPD [0.922,0.958] [0.911,0.953] [0.917,0.956] [0.906,0.950]
IF (NSE) 7.2 (0.0003) 9.2 (0.0004) 7.5 (0.0003) 9.1 (0.0003)
Table 5.10: Summary of the posterior sample of the persistence of log volatility, φ, in the LR-NSV-NCT and LR-NSV-SKT models for the TOPIX 2004–2007 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
LR-NSV-NCT model φET
Mean (SD) 0.940 (0.011) 0.932 (0.012) 0.937 (0.011) 0.929 (0.013) 90% HPD [0.923,0.959] [0.912,0.954] [0.917,0.957] [0.906,0.951]
IF (NSE) 6.2 (0.0003) 8.2 (0.0003) 6.6 (0.0003) 9.5 (0.0005) φM T
Mean (SD) 0.955 (0.025) 0.954 (0.025) 0.955 (0.026) 0.936 (0.028) 90% HPD [0.919,0.996] [0.909,0.995] [0.918,0.995] [0.892,0.986]
IF (NSE) 28.0 (0.0013) 22.5 (0.0011) 23.2 (0.0012) 23.0 (0.0013) φY J
Mean (SD) 0.939 (0.011) 0.930 (0.013) 0.937 (0.012) 0.929 (0.013) 90% HPD [0.921,0.959] [0.910,0.954] [0.916,0.956] [0.907,0.950]
IF (NSE) 8.1 (0.0004) 10.3 (0.0004) 7.4 (0.0004) 7.9 (0.0004) LR-NSV-SKT model
φET
Mean (SD) 0.940 (0.011) 0.932 (0.012) 0.938 (0.012) 0.930 (0.013) 90% HPD [0.922,0.960] [0.912,0.953] [0.918,0.958] [0.909,0.952]
IF (NSE) 7.6 (0.0003) 8.5 (0.0004) 7.7 (0.0004) 9.2 (0.0003) φM T
Mean (SD) 0.951 (0.025) 0.944 (0.027) 0.956 (0.024) 0.935 (0.027) 90% HPD [0.913,0.995] [0.899,0.990] [0.921,0.998] [0.891,0.982]
IF (NSE) 26.5 (0.0014) 26.4 (0.0015) 24.0 (0.0009) 21.1 (0.0010) φY J
Mean (SD) 0.939 (0.011) 0.931 (0.013) 0.937 (0.012) 0.929 (0.013) 90% HPD [0.920,0.958] [0.910,0.953] [0.916,0.956] [0.909,0.951]
IF (NSE) 6.8 (0.0003) 7.2 (0.0003) 8.4 (0.0004) 7.6 (0.0003)
Table 5.11: Summary of the posterior sample of the persistence of log volatility, φ, in the LR-NSV and LR-NSV-T models for the TOPIX 2004–2011 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
LR-NSV model φET
Mean (SD) 0.947 (0.008) 0.950 (0.007) 0.946 (0.008) 0.950 (0.007) 90% HPD [0.933,0.961] [0.937,0.963] [0.932,0.960] [0.937,0.962]
IF (NSE) 6.9 (0.0002) 6.6 (0.0001) 5.1 (0.0002) 6.6 (0.0002) φM T
Mean (SD) 0.927 (0.015) 0.926 (0.017) 0.931 (0.016) 0.925 (0.016) 90% HPD [0.902,0.953] [0.897,0.953] [0.905,0.957] [0.898,0.953]
IF (NSE) 13.8 (0.0004) 19.1 (0.0007) 15.5 (0.0006) 15.6 (0.0005) φY J
Mean (SD) 0.947 (0.008) 0.950 (0.007) 0.946 (0.008) 0.950 (0.007) 90% HPD [0.933,0.960] [0.937,0.963] [0.932,0.960] [0.937,0.963]
IF (NSE) 6.8 (0.0002) 6.5 (0.0001) 5.9 (0.0002) 6.1 (0.0002) LR-NSV-T model
φET
Mean (SD) 0.948 (0.008) 0.951 (0.007) 0.947 (0.008) 0.951 (0.007) 90% HPD [0.934,0.961] [0.938,0.963] [0.934,0.961] [0.939,0.963]
IF (NSE) 5.3 (0.0001) 9.2 (0.0002) 5.9 (0.0002) 5.3 (0.0002) φM T
Mean (SD) 0.927 (0.016) 0.929 (0.016) 0.932 (0.015) 0.925 (0.016) 90% HPD [0.900,0.955] [0.901,0.956] [0.906,0.958] [0.897,0.952]
IF (NSE) 19.1 (0.0007) 16.8 (0.0006) 15.1 (0.0006) 16.9 (0.0006) φY J
Mean (SD) 0.949 (0.007) 0.951 (0.007) 0.947 (0.008) 0.951 (0.007) 90% HPD [0.935,0.961] [0.939,0.963] [0.933,0.961] [0.939,0.963]
IF (NSE) 5.5 (0.0002) 7.5 (0.0002) 6.2 (0.0002) 7.5 (0.0002)
Table 5.12: Summary of the posterior sample of the persistence of log volatility, φ, in the LR-NSV-NCT and LR-NSV-SKT models for the TOPIX 2004–2011 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
LR-NSV-NCT model φET
Mean (SD) 0.947 (0.008) 0.950 (0.007) 0.946 (0.008) 0.950 (0.007) 90% HPD [0.934,0.961] [0.938,0.963] [0.933,0.960] [0.932,0.960]
IF (NSE) 6.6 (0.0002) 5.9 (0.0002) 5.8 (0.0002) 6.2 (0.0002) φM T
Mean (SD) 0.927 (0.016) 0.928 (0.016) 0.932 (0.016) 0.924 (0.016) 90% HPD [0.900,0.953] [0.900,0.955] [0.906,0.958] [0.898,0.953]
IF (NSE) 16.6 (0.0007) 17.9 (0.0007) 14.6 (0.0005) 20.3 (0.0006) φY J
Mean (SD) 0.948 (0.008) 0.951 (0.007) 0.947 (0.008) 0.951 (0.007) 90% HPD [0.935,0.961] [0.940,0.964] [0.934,0.961] [0.939,0.964]
IF (NSE) 6.6 (0.0002) 5.6 (0.0002) 5.6 (0.0002) 6.3 (0.0002) LR-NSV-SKT model
φET
Mean (SD) 0.948 (0.007) 0.952 (0.007) 0.947 (0.008) 0.951 (0.007) 90% HPD [0.936,0.962] [0.940,0.964] [0.933,0.960] [0.940,0.964]
IF (NSE) 6.8 (0.0002) 6.4 (0.0002) 5.2 (0.0001) 6.9 (0.0002) φM T
Mean (SD) 0.927 (0.016) 0.927 (0.016) 0.932 (0.016) 0.924 (0.017) 90% HPD [0.902,0.954] [0.898,0.955] [0.904,0.958] [0.896,0.952]
IF (NSE) 14.8 (0.0006) 19.5 (0.0007) 15.0 (0.0005) 16.9 (0.0007) φY J
Mean (SD) 0.949 (0.007) 0.952 (0.007) 0.948 (0.008) 0.952 (0.007) 90% HPD [0.936,0.962] [0.940,0.964] [0.934,0.961] [0.940,0.964]
IF (NSE) 6.5 (0.0002) 7.5 (0.0002) 5.7 (0.0002) 6.7 (0.0002)
General Conclusion
This dissertation contains three chapters that aim to contribute to the literature related to the estimation of return volatility using the leveraged RSV model.
In the third chapter, we have made a contribution to the literature of MCMC methods for estimating the leveraged SV model. We explicitly extend the HMC and RMHMC methods to the leveraged SV model. We compared the computa-tional efficiency of these methods with the multi-move Metropolis-Hastings method for drawing latent volatility. The RMHMC method is most efficient for sampling parameters and very highly efficient for sampling latent volatility with very high acceptance rate for latent volatility.
In the fourth chapter, we have contributed to the extension of leveraged RSV model. We present an extension which allows capturing both flexible skewness and heavy-tailedness in returns instead of heavy-tailedness or normality. We also allow flexible persistence in RV suggesting a varying persistence when the RV data sampled at very high frequency (say 1-minute). The results from our empirical application using stock indices also demonstrated that the leveraged RSV model with SKT distribution provide the best fit to the TOPIX data, followed by the model with NCT distribution in terms of Bayes factor. In particular, the model with SKT distribution are very strongly favored against the competing models. An efficient RMHMC sampling procedure was developed for estimating the extended models.
Finally, in the fifth chapter, we have contributed to extend the leveraged RSV model with generalized Student’s t-distribution to the power transformation of lagged volatility process. We apply three families of power transformation—
exponential, modulus, and Yeo–Johnson—to transform lag volatility. These fami-lies permit transformed data to be non-positive and include linear case. After per-formance evaluation, we conclude that the non-linear specification outperforms the linear specification for any RV measures. We also find that the non-linear model with SKT distribution performs the best among four returns distributions. We highlight the importance of using modulus transformation that are robust to the returns data having a very high kurtosis. In contrast, this transformation worst
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fitted the lagged volatility for the returns data having a small kurtosis. Addition-ally, the LR-MTSV-SKT model’s performance showed considerable robustness for priors with very diffused distributional behaviour. An efficient HMC sampling procedure was developed for estimating the latent volatility and power parameter.
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